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Student[LinearAlgebra]

 EigenPlot
 apply a linear transformation, highlighting the eigenvectors

 Calling Sequence EigenPlot(M, opts)

Parameters

 M - 2x2 or 3x3 Matrix opts - plotting options or equation(s) of the form option = value where option is one of showcircle, circleoptions, showeigenvectors, eigenoptions, showimages, imageoptions, showunitvectors, unitoptions, showsphere, sphereoptions, eigencolors, numvectors, region, spheregrid, or Student plot options; specify options for the plot

Description

 • The EigenPlot(M) produces a plot showing the (real) eigenvectors of the Matrix M.  This plot has the following components.
 * The images of a selection of unit vectors emanating from the origin; these images are drawn with their bases located at the tips of the corresponding unit vectors.
 * The unit circle (2-D) or unit sphere (3-D).
 * The (real) eigenvectors; in the 2-D case, these vectors are drawn with their bases located on the unit circle.  In the 3-D case, they are drawn with their bases on the unit sphere if they point outward, and otherwise they are drawn with their heads on the unit sphere.  Note that the eigenvectors are drawn in pairs pointing in opposite directions.  Because these eigenvectors are images of unit vectors, their lengths are determined by the corresponding eigenvalues. Thin lines are drawn as diameters of the unit circle or sphere, which are extensions of the eigenvectors. These diameter lines provide additional information about the corresponding eigenspaces, which is particularly useful in the case of eigenvalues that are $0$.
 • The opts argument can include any of the Student plot options or any of the following equations that set plot options.
 showcircle = true or false
 In the 2-D case, select whether to plot the unit circle.  This option is ignored in the 3-D case. [Default: true]
 circleoptions = list
 A list of options which control the display of the unit circle. These options are passed to plot.
 showeigenvectors = true or false
 Select whether to plot the eigenvectors.  [Default: true]
 eigenoptions = list
 A list of options which control the display of the eigenvectors.  Because the eigenvectors are plotted using the plots[arrow] command, only corresponding options are allowed.
 showimagevectors = true or false
 Select whether to display the images of the unit vectors under the action of the Matrix M.  [Default: true]
 imageoptions = list
 A list of options which control the display of the images of the unit vectors under the action of the Matrix M.  Because these image vectors are plotted using the plots[arrow] command, only corresponding options are allowed.
 Note: By default, the image vectors for a 2-D plot are drawn using the the double_arrow style, and the image vectors for a 3-D plot are drawn using the harpoon style to enable them to be visually distinguished from the eigenvectors.
 showsphere = true or false
 In the 3-D case, select whether to plot the unit sphere.  This option is ignored in the 2-D case. [Default: true]
 sphereoptions = list
 A list of options which control the display of the unit sphere.  These options are passed to plot3d.
 showunitvectors = true or false
 For the 2-D case only, select whether to display the unit vectors (these are the vectors to which the Matrix is applied to produce the image vectors). [Default:  false]
 unitoptions = list
 A list of options which control the display of the unit vectors.  Because these unit vectors are plotted using the plots[arrow] command, only corresponding options are allowed.  (These vectors are not available in the 3-D case.)
 eigencolors = list
 A list of 2 (2-D) or 3 (3-D) colors to be used when plotting the eigenvectors. [Default: [red, blue], [red, blue, green]]
 numvectors = posint
 Specify the number of unit and/or image vectors to display.
 Note: In the 3-D case, if the region option is included, and specifies a region other than sphere, the actual number of image vectors drawn may be less than the value specified in this numvectors option. [Default: 21 (2-D), 25 (3-D), 15 (3-D, octant region)]
 region = sphere, octant, righthalf, lefthalf, fronthalf, backhalf, tophalf, bottomhalf
 For a 3-D problem (that is, a 3 x 3 Matrix M), select how much of the unit sphere to display.  [Default: sphere]
 spheregrid = list of 2 positive integers
 For a 3-D problem, specify the grid used to draw the unit sphere.  [Default: [15,15]]
 caption = anything
 A caption for the plot.
 The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right):$
 > $\mathrm{infolevel}\left[\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right]≔1:$
 > $M≔⟨⟨1,2⟩|⟨1,-1⟩⟩:$
 > $\mathrm{EigenPlot}\left(M\right)$
 Eigenvalue: 3^(1/2) Multiplicity: 1 Eigenvector: < 1.366, 1. > Eigenvalue: -3^(1/2) Multiplicity: 1 Eigenvector: < -.3660, 1. > > $M≔⟨⟨2,0,0⟩|⟨-1,-1,-4⟩|⟨-1,2,5⟩⟩$
 ${M}{≔}\left[\begin{array}{ccc}{2}& {-1}& {-1}\\ {0}& {-1}& {2}\\ {0}& {-4}& {5}\end{array}\right]$ (1)
 > $\mathrm{EigenPlot}\left(M\right)$
 Eigenvalue: 3 Multiplicity: 1 Eigenvector: < -3/2, 1/2, 1 > Eigenvalue: 1 Multiplicity: 1 Eigenvector: < 2, 1, 1 > Eigenvalue: 2 Multiplicity: 1 Eigenvector: < 1, 0, 0 > > $\mathrm{EigenPlot}\left(M,\mathrm{showsphere}=\mathrm{false},\mathrm{showimagevectors}=\mathrm{false},\mathrm{eigenoptions}=\left[\mathrm{shape}=\mathrm{harpoon}\right]\right)$
 Eigenvalue: 3 Multiplicity: 1 Eigenvector: < -3/2, 1/2, 1 > Eigenvalue: 1 Multiplicity: 1 Eigenvector: < 2, 1, 1 > Eigenvalue: 2 Multiplicity: 1 Eigenvector: < 1, 0, 0 > The commands to create the plots from the Plotting Guide are

 > $M≔⟨⟨1,1⟩|⟨1,2⟩⟩:$
 > $\mathrm{EigenPlot}\left(M,\mathrm{eigenoptions}=\left[\mathrm{color}=\mathrm{black}\right]\right)$
 Eigenvalue: 3/2+1/2*5^(1/2) Multiplicity: 1 Eigenvector: < .6180, 1. > Eigenvalue: 3/2-1/2*5^(1/2) Multiplicity: 1 Eigenvector: < -1.618, 1. > > $M≔⟨⟨1,2,3⟩|⟨2,0,1⟩|⟨1,-1,0⟩⟩:$
 > $\mathrm{EigenPlot}\left(M\right)$
 Eigenvalue: 1/6*(-100+36*I*331^(1/2))^(1/3)+38/3/(-100+36*I*331^(1/2))^(1/3)+1/3 Multiplicity: 1 Eigenvector: < .8499, .2532-.5082e-2*I, 1. > Eigenvalue: -1/12*(-100+36*I*331^(1/2))^(1/3)-19/3/(-100+36*I*331^(1/2))^(1/3)+1/3+1/2*I*3^(1/2)*(1/6*(-100+36*I*331^(1/2))^(1/3)-38/3/(-100+36*I*331^(1/2))^(1/3)) Multiplicity: 1 Eigenvector: < -1.282, 1.585, 1. > Eigenvalue: -1/12*(-100+36*I*331^(1/2))^(1/3)-19/3/(-100+36*I*331^(1/2))^(1/3)+1/3-1/2*I*3^(1/2)*(1/6*(-100+36*I*331^(1/2))^(1/3)-38/3/(-100+36*I*331^(1/2))^(1/3)) Multiplicity: 1 Eigenvector: < .3573, -.5930, 1. > 